FINANCIAL SIGNAL PROCESSING AND MACHINE LEARNING FINANCIAL SIGNAL PROCESSING AND MACHINE LEARNING Edited by Ali N Akansu New Jersey Institute of Technology, USA Sanjeev R Kulkarni Princeton University, USA Dmitry Malioutov IBM T.J Watson Research Center, USA This edition first published 2016 © 2016 John Wiley & Sons, Ltd First Edition published in 2016 Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data applied for ISBN: 9781118745670 A catalogue record for this book is available from the British Library Set in 10/12pt, TimesLTStd by SPi Global, Chennai, India 2016 Contents List of Contributors Preface 1.1 1.2 1.3 1.4 Overview Ali N Akansu, Sanjeev R Kulkarni, and Dmitry Malioutov Introduction A Bird’s-Eye View of Finance 1.2.1 Trading and Exchanges 1.2.2 Technical Themes in the Book Overview of the Chapters 1.3.1 Chapter 2: “Sparse Markowitz Portfolios” by Christine De Mol 1.3.2 Chapter 3: “Mean-Reverting Portfolios: Tradeoffs between Sparsity and Volatility” by Marco Cuturi and Alexandre d’Aspremont 1.3.3 Chapter 4: “Temporal Causal Modeling” by Prabhanjan Kambadur, Aurélie C Lozano, and Ronny Luss 1.3.4 Chapter 5: “Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process” by Mustafa U Torun, Onur Yilmaz and Ali N Akansu 1.3.5 Chapter 6: “Approaches to High-Dimensional Covariance and Precision Matrix Estimation” by Jianqing Fan, Yuan Liao, and Han Liu 1.3.6 Chapter 7: “Stochastic Volatility: Modeling and Asymptotic Approaches to Option Pricing and Portfolio Selection” by Matthew Lorig and Ronnie Sircar 1.3.7 Chapter 8: “Statistical Measures of Dependence for Financial Data” by David S Matteson, Nicholas A James, and William B Nicholson 1.3.8 Chapter 9: “Correlated Poisson Processes and Their Applications in Financial Modeling” by Alexander Kreinin 1.3.9 Chapter 10: “CVaR Minimizations in Support Vector Machines” by Junya Gotoh and Akiko Takeda 1.3.10 Chapter 11: “Regression Models in Risk Management” by Stan Uryasev Other Topics in Financial Signal Processing and Machine Learning References xiii xv 1 6 7 7 8 8 9 Contents vi 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 Sparse Markowitz Portfolios Christine De Mol Markowitz Portfolios Portfolio Optimization as an Inverse Problem: The Need for Regularization Sparse Portfolios Empirical Validation Variations on the Theme 2.5.1 Portfolio Rebalancing 2.5.2 Portfolio Replication or Index Tracking 2.5.3 Other Penalties and Portfolio Norms Optimal Forecast Combination Acknowlegments References 11 Mean-Reverting Portfolios Marco Cuturi and Alexandre d’Aspremont Introduction 3.1.1 Synthetic Mean-Reverting Baskets 3.1.2 Mean-Reverting Baskets with Sufficient Volatility and Sparsity Proxies for Mean Reversion 3.2.1 Related Work and Problem Setting 3.2.2 Predictability 3.2.3 Portmanteau Criterion 3.2.4 Crossing Statistics Optimal Baskets 3.3.1 Minimizing Predictability 3.3.2 Minimizing the Portmanteau Statistic 3.3.3 Minimizing the Crossing Statistic Semidefinite Relaxations and Sparse Components 3.4.1 A Semidefinite Programming Approach to Basket Estimation 3.4.2 Predictability 3.4.3 Portmanteau 3.4.4 Crossing Stats Numerical Experiments 3.5.1 Historical Data 3.5.2 Mean-reverting Basket Estimators 3.5.3 Jurek and Yang (2007) Trading Strategy 3.5.4 Transaction Costs 3.5.5 Experimental Setup 3.5.6 Results Conclusion References 23 11 13 15 17 18 18 19 19 20 21 21 23 24 24 25 25 26 27 28 28 29 29 29 30 30 30 31 31 32 32 33 33 33 36 36 39 39 Contents vii 41 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 Temporal Causal Modeling Prabhanjan Kambadur, Aurélie C Lozano, and Ronny Luss Introduction TCM 4.2.1 Granger Causality and Temporal Causal Modeling 4.2.2 Grouped Temporal Causal Modeling Method 4.2.3 Synthetic Experiments Causal Strength Modeling Quantile TCM (Q-TCM) 4.4.1 Modifying Group OMP for Quantile Loss 4.4.2 Experiments TCM with Regime Change Identification 4.5.1 Model 4.5.2 Algorithm 4.5.3 Synthetic Experiments 4.5.4 Application: Analyzing Stock Returns Conclusions References Explicit Kernel and Sparsity of Eigen Subspace for the AR(1) Process Mustafa U Torun, Onur Yilmaz, and Ali N Akansu Introduction Mathematical Definitions 5.2.1 Discrete AR(1) Stochastic Signal Model 5.2.2 Orthogonal Subspace Derivation of Explicit KLT Kernel for a Discrete AR(1) Process 5.3.1 A Simple Method for Explicit Solution of a Transcendental Equation 5.3.2 Continuous Process with Exponential Autocorrelation 5.3.3 Eigenanalysis of a Discrete AR(1) Process 5.3.4 Fast Derivation of KLT Kernel for an AR(1) Process Sparsity of Eigen Subspace 5.4.1 Overview of Sparsity Methods 5.4.2 pdf-Optimized Midtread Quantizer 5.4.3 Quantization of Eigen Subspace 5.4.4 pdf of Eigenvector 5.4.5 Sparse KLT Method 5.4.6 Sparsity Performance Conclusions References 41 46 46 47 49 51 52 52 53 55 56 58 60 62 63 64 67 67 68 68 69 72 73 74 76 79 82 83 84 86 87 89 91 97 97 Contents viii 6.1 6.2 6.3 6.4 6.5 6.6 Approaches to High-Dimensional Covariance and Precision Matrix Estimations Jianqing Fan, Yuan Liao, and Han Liu Introduction Covariance Estimation via Factor Analysis 6.2.1 Known Factors 6.2.2 Unknown Factors 6.2.3 Choosing the Threshold 6.2.4 Asymptotic Results 6.2.5 A Numerical Illustration Precision Matrix Estimation and Graphical Models 6.3.1 Column-wise Precision Matrix Estimation 6.3.2 The Need for Tuning-insensitive Procedures 6.3.3 TIGER: A Tuning-insensitive Approach for Optimal Precision Matrix Estimation 6.3.4 Computation 6.3.5 Theoretical Properties of TIGER 6.3.6 Applications to Modeling Stock Returns 6.3.7 Applications to Genomic Network Financial Applications 6.4.1 Estimating Risks of Large Portfolios 6.4.2 Large Panel Test of Factor Pricing Models Statistical Inference in Panel Data Models 6.5.1 Efficient Estimation in Pure Factor Models 6.5.2 Panel Data Model with Interactive Effects 6.5.3 Numerical Illustrations Conclusions References 100 100 101 103 104 105 105 107 109 110 111 112 114 114 115 118 119 119 121 126 126 127 130 131 131 Stochastic Volatility Matthew Lorig and Ronnie Sircar 135 7.1 Introduction 7.1.1 Options and Implied Volatility 7.1.2 Volatility Modeling Asymptotic Regimes and Approximations 7.2.1 Contract Asymptotics 7.2.2 Model Asymptotics 7.2.3 Implied Volatility Asymptotics 7.2.4 Tractable Models 7.2.5 Model Coefficient Polynomial Expansions 7.2.6 Small “Vol of Vol” Expansion 7.2.7 Separation of Timescales Approach 7.2.8 Comparison of the Expansion Schemes Merton Problem with Stochastic Volatility: Model Coefficient Polynomial Expansions 135 136 137 141 142 142 143 145 146 152 152 154 7.2 7.3 155 Financial Signal Processing and Machine Learning 282 However, in this form, they are rarely used in practice If X1 , … , Xn and Y are discretely distributed with joint probability distribution ∑m ℙ[X1 = x1j , …, Xn = xnj , Y = yj ] = pj > 0, j = 1, … , m, where j=1 pj = 1, then with the formula (5) in Rockafellar et al (2006a), the quantile regression (11.37) can be restated as the linear program ( ) n m ∑ ∑ pj yj − c0 + 𝛼 −1 𝜁j − ck xkj c0 , c1 , … , cn j=1 k=1 𝜁1 , … , 𝜁 m subject to 𝜁j ≥ c0 + n ∑ ck xkj − yj , 𝜁j ≥ 0, j = 1, … , m, (11.39) k=1 where 𝜁1 , … , 𝜁m are auxiliary variables The return-based style classification for a mutual fund is a regression of the fund return on several indices as explanatory variables, where regression coefficients represent the fund’s style with respect to each of the indices In contrast to the least-squares regression, the quantile regression can assess the impact of explanatory variables on various parts of the regressand distribution, for example on the 95th and 99th percentiles Moreover, for a portfolio with exposure to derivatives, the mean and quantiles of the portfolio return distribution may have quite different regression coefficients for the same explanatory variables For example, in most cases, the strategy of investing into naked deep out-of-the-money options behaves like a bond paying some interest; however, in some rare cases, this strategy may lose significant amounts of money With the quantile regression, a fund manager can analyze the impact of a particular factor on any part of the return distribution Example 11.11 presents an unconstrained quantile regression problem arising in the return-based style classification of a mutual fund Example 11.11 (quantile regression in style classification) Let L(c, X) be a loss function that linearly depends on a decision vector c = (c1 , … , cn ) and on a random vector X = (X1 , … , Xn ) representing uncertain rates of return of n indices as explanatory variables The quantile regression (11.37) with L(c, X) in place of Z takes the form E[𝛼[L(c, X)]+ + (1 − 𝛼)[L(c, X)]− ] (11.40) c1 ,…,cn A constrained quantile regression is formulated similarly to (11.37): c0 ,c1 ,…,cn E[𝛼 Z+ + (1 − 𝛼) Z− ] with Z = Y − c0 − subject to (c0 , c1 , … , cn ) ∈ C, n ∑ ck Xk k=1 (11.41) where C is a given feasible set for regression coefficients c0 , c1 , … , cn Example 11.12 (index tracking with asymmetric mean absolute error) The setting is identical to that in Example 11.2 But this time, the allocation positions c1 , … , cn are found from the constrained quantile regression (11.41) with C given by (11.25) If X1 , … , Xn and Regression Models in Risk Management 283 Y are assumed to be discretely distributed with joint probability distribution ℙ[X1 = x1j , …, ∑ Xn = xnj , Y = yj ] = pj > 0, j = 1, … , m, where m j=1 pj = 1, then this regression problem can be formulated as the linear program ( ) n m ∑ ∑ −1 pj yj − ck xkj + 𝛼 𝜁j c1 , … , cn j=1 k=1 𝜁1 , … , 𝜁 m subject to 𝜁j ≥ n ∑ ck xkj − yj , 𝜁j ≥ 0, j = 1, … , m, k=1 n ∑ ck = 1, ck ≥ 0, k = 1, … , n, k=1 where 𝜁1 , … , 𝜁m are auxiliary variables The linear regression with the mixed quantile error measure (11.9) is called mixed quantile regression It generalizes quantile regression and, through error decomoposition, takes the form [ [ { ] ( }] ) n l n ∑ ∑ ∑ cj Xj + 𝜆k E max 0, Ck − cj Xj − Ck E Y − 𝛼k c ,…,c , j=1 k=1 j=1 n C1 , … , Cl (11.42) with the intercept c0 determined by c0 = l ∑ 𝜆k Ck , k=1 where C1 , … , Cl are a solution to (11.42); see Example 3.1 in Rockafellar et al (2008) The optimality conditions (11.20) for (11.42) are complicated However, as the quantile regression, (11.42) can be reduced to a linear program 11.8 Special Types of Linear Regression This section discusses special types of unconstrained and constrained linear regressions encountered in statistical decision problems Often, it is required to find an unbiased linear approximation of an output random variable Y by a linear combination of input random variables X1 , … , Xn , in which case the approximation ∑ error has zero expected value: E[Y − c0 − nk=1 ck Xk ] = A classical example of an unbiased linear regression is minimizing variance or, equivalently, standard deviation with the intercept ∑ c0 set to c0 = E[Y − nk=1 ck Xk ] If, in this example, the standard deviation is replaced by a general deviation measure D, we obtain a generalized unbiased linear regression: ̃ and D(Z) c1 ,…,cn ̃ c0 = E[Z], where ̃=Y− Z n ∑ k=1 ck Xk (11.43) Financial Signal Processing and Machine Learning 284 In fact, (11.43) is equivalent to minimizing the error measure E(Z) = D(Z) + |E[Z]| of ∑ Z = Y − c0 − nk=1 ck Xk Observe that in view of the error decomposition theorem (Rockafellar et al., 2008, Theorem 3.2), the generalized linear regression (11.18) with a nondegenerate error measure E and the unbiased linear regression (11.43) with the deviation measure D projected from E yield the same c1 , … , cn but, in general, different intercepts c0 Rockafellar et al (2008) introduced risk acceptable linear regression in which a devia∑ tion measure D of the approximation error Z = Y − c0 − nk=1 ck Xk is minimized subject to a constraint on the averse measure of risk R related to D by R(X) = D(X) − E[X]: c1 ,…,cn D(Z) subject to R(Z) = with Z = Y − c0 − n ∑ ck Xk , (11.44) k=1 which is equivalent to ̃ and D(Z) c1 ,…,cn ̃ − D(Z), ̃ c0 = E(Z) where ̃=Y− Z n ∑ ck Xk (11.45) k=1 The unbiased linear regression (11.43) and risk acceptable linear regression (11.45) show that the intercept c0 could be set based on different requirements In general, the risk acceptable regression may minimize either an error measure E or a deviation measure D of the error Z subject to a constraint on a risk measure R of Z not necessarily related to E or D Example 11.13 illustrates a risk acceptable regression arising in a portfolio replication problem with a constraint on CVaR Example 11.13 (risk acceptable regression) Let L(c, X) be a portfolio replication error (loss function) that linearly depends on a decision vector c = (c1 , … , cn ) and on a random vector X = (X1 , … , Xn ) representing uncertain rates of return of n instruments in a portfolio replicating the S&P 100 index The risk acceptable regression minimizes the mean absolute error ∑ of L(c, X) subject to the budget constraint ni=1 ci ≤ U with known a1 , … , an and U and subject to a CVaR constraint on the underperformance of the portfolio compared to the index: ‖L(c, X)‖1 c1 ,…,cn subject to n ∑ ci ≤ U, i=1 CVaR𝛼 (L(c, X)) ≤ 𝑤, ci ≥ 0, i = 1, … , n, (11.46) where 𝛼 and 𝑤 are given 11.9 Robust Regression Robust regression aims to reduce influence of sample outliers on regression parameters, especially when regression error has heavy tails In statistics, robustness of an estimator is a well-established notion and is assessed by the so-called estimator’s breakdown point, which is the proportion of additional arbitrarily large Regression Models in Risk Management 285 observations (outliers) needed to make the estimator unbounded For example, the sample mean requires just a single such observation, while the sample median would still be finite until the proportion of such observations reaches 50% Consequently, the mean’s breakdown point is 0%, whereas the median’s breakdown point is 50% As in the previous regression setting, suppose Y is approximated by a linear combination of ∑ input random variables X1 , … , Xn with the regression error defined by Z = Y − c0 − ni=1 ci Xi , where c0 , c1 , … , cn are unknown regression coefficients A robust regression minimizes an error measure of Z that has a nonzero breakdown point Thus, in this setting, the regression’s breakdown point is that of the error measure Often, a robust regression relies on order statistics of Z and on “trimmed” error measures Two popular robust regressions are the least median of squares (LMS) regression, which minimizes the median of Z and has a 50% breakdown point: c0 ,c1 ,…,cn med (Z ) with Z = Y − c0 − n ∑ ci Xi (11.47) i=1 and the least-trimmed-squares (LTS) regression, which minimizes the average 𝛼-quantile of Z and has a (1 − 𝛼)⋅100% breakdown point: q̄ (𝛼) c0 ,c1 ,…,cn Z with Z = Y − c0 − n ∑ ci Xi (11.48) i=1 Rousseeuw and Driessen (2006) referred to (11.48) as a challenging optimization problem Typically, in the LTS regression, 𝛼 is set to be slightly larger than 1∕2 For 𝛼 = 1, q̄ Z (𝛼) = ‖Z‖22 , and (11.48) reduces to the standard least-squares regression The LTS regression is reported to have advantages over the LMS regression or the one that minimizes the 𝛼-quantile of Z ; see Rousseeuw and Driessen (2006), Rousseeuw and Leroy (1987), and Venables and Ripley (2002) Let h be such that h(t) > for t ≠ and h(0) = 0, but not necessarily symmetric (i.e., h(−t) ≠ h(t) in general) Then, the LMS and LTS regressions have the following generalization: Minimizing the upper 𝛼-quantile of h(Z): c0 ,c1 ,…,cn q+h(Z) (𝛼) with Z = Y − c0 − n ∑ ci Xi , (11.49) ci Xi (11.50) i=1 Minimizing the average 𝛼-quantile of h(Z): c0 ,c1 ,…,cn q̄ h(Z) (𝛼) with Z = Y − c0 − n ∑ i=1 For example, in both (11.49) and (11.50), we may use h(Z) = |Z|p , p ≥ In particular, for h(Z) = Z , (11.49) with 𝛼 = 1∕2 corresponds to the LMS regression (11.47), whereas (11.50) reduces to the LTS regression (11.48) When h(−t) = h(t), (11.49) and (11.50) not discriminate positive and negative errors This, however, is unlikely to be appropriate for errors with significantly skewed distributions Financial Signal Processing and Machine Learning 286 For example, instead of med (Z ) and q̄ Z (𝛼), we can use two-tailed 𝛼-value-at-risk (VaR) deviation of the error Z defined by TwoTailVaRΔ 𝛼 (Z) = VaR1−𝛼 (Z) + VaR1−𝛼 (−Z) ≡ q−Z (𝛼) − q+Z (1 − 𝛼), 𝛼 ∈ (1∕2, 1] (11.51) The definition (11.51) shows that the two-tailed 𝛼-VaR deviation is, in fact, the range between the upper and lower (1 − 𝛼)-tails of the error Z, which is equivalent to the support of the random variable Z with truncated (1 − 𝛼)⋅100% of the “outperformances” and (1 − 𝛼)⋅100% of “underperformances.” Consequently, the two-tailed 𝛼-VaR deviation has the breakdown point of (1 − 𝛼)⋅100% Typically, 𝛼 is chosen to be 0.75 and 0.9 Robust regression is used in mortgage pipeline hedging Usually, mortgage lenders sell mortgages in the secondary market Alternatively, they can exchange mortgages for mortgage-backed securities (MBSs) and then sell MBSs in the secondary market The mortgage-underwriting process is known as the “pipeline.” Mortgage lenders commit to a mortgage interest rate while the loan is in process, typically for a period of 30–60 days If the rate rises before the loan goes to closing, the value of the loan declines and the lender sells the loan at a lower price The risk that mortgages in process will fall in value prior to their sale is known as mortgage pipeline risk Lenders often hedge this exposure either by selling forward their expected closing volume or by shorting either US Treasury notes or futures contracts Fallout refers to the percentage of loan commitments that not go to closing It affects the mortgage pipeline risk As interest rates fall, the fallout rises because borrowers locked in a mortgage rate are more likely to find better rates with another lender Conversely, as rates rise, the percentage of loans that close increases So, the fallout alters the size of the pipeline position to be hedged against and, as a result, affects the required size of the hedging instrument: at lower rates, fewer rate loans will close and a smaller position in the hedging instrument is needed To hedge against the fallout risk, lenders often use options on US Treasury note futures Suppose a hedging portfolio is formed out of n hedging instruments with random returns X1 , … , Xn A pipeline risk hedging problem is to minimize a deviation measure D of the underperformance of the hedging portfolio with respect to a random hedging target Y, where short sales are allowed and transaction costs are ignored Example 11.14 formulates a robust regression with the two-tailed 𝛼-VaR deviation used in a mortgage pipeline hedging problem Example 11.14 (robust regression with two-tailed 𝜶-VaR deviation) Let a target random variable Y be approximated by a linear combination of n random variables X1 , … , Xn , then the ∑ robust regression minimizes the two-tailed 𝛼-VaR deviation of the error Y − c0 − ni=1 ci Xi : ( ) n ∑ Δ ci Xi TwoTailVaR𝛼 Y − c0 − (11.52) c0 ,c1 ,…,cn It has a (1 − 𝛼)⋅100%-breakdown point i=1 Regression Models in Risk Management 287 References, Further Reading, and Bibliography Hastie, T., Tibshirani, R and Friedman, J (2008) The elements of statistical learning: data mining, inference, and prediction, 2nd ed New York: Springer Koenker, R and Bassett, G (1978) Regression quantiles Econometrica, 46, 33–50 Men, C., Romeijn, E., Taskin, C and Dempsey, J (2008) An exact approach to direct aperture optimization in IMRT treatment planning Physics in Medicine and Biology, 52, 7333–7352 Rockafellar, R.T., Uryasev, S and Zabarankin, M (2002) Deviation measures in risk analysis and optimization Technical Report 2002-7, ISE Department, University of Florida, Gainesville, FL Rockafellar, R.T., Uryasev, S and Zabarankin, M (2006a) Generalized deviations in risk analysis Finance and Stochastics, 10, 51–74 Rockafellar, R.T., Uryasev, S and Zabarankin, M (2006b) Optimality conditions in portfolio analysis with general deviation measures Mathematical Programming, Series B, 108, 515–540 Rockafellar, R.T., Uryasev, S and Zabarankin, M (2008) Risk tuning with generalized linear regression Mathematics of Operations Research, 33, 712–729 Rousseeuw, P.J and Driessen, K (2006) Computing LTS regression for large data sets Data Mining and Knowledge Discovery, 12, 29–45 Rousseeuw, P and Leroy, A (1987) Robust regression and outlier detection New York: John Wiley & Sons van der Waerden, B (1957) Mathematische Statistik Berlin: Springer-Verlag Venables, W and Ripley, B (2002) Modern applied statistics with S-PLUS, 4th ed New York: Springer Index References to figures are given in bold type References to tables are given in italic type ACF see autocorrelation functions adaptive lasso, 48, 50, 51 anti-monotone sets, 208 AR(1) process Eigen subspace, 70, 91–2 KLT kernel derivation, 79–82 orthogonal subspace, 69–71 Karhunen-Loeve transform, 70–1 performance matrices, 71 power spectral density, 78 stochastic signal model, 68 arbitrage, ARCH model, 141 nonparametric tests, 173–4 parametric tests, 172–3 Archimedean copula, 181, 182 assets, mean-reverting, 23–4 autocorrelation functions (ACF), 164–5 Kendall’s tau, 166–7, 169 misspecification testing conditional heteroscedascicity, 171–2 Ljung-Box, 171 Spearman’s rho, 168–9, 169–71 autocovariance, 164 autoregressive conditional heteroscedasticity (ARCH), 141 backward simulation (BS), 222–7 BEKK Garch model, 175 bias, 267 BIC criterion, 50 bid-ask price, big data finance, Black-Scholes model, 136–7, 203 BS method, 203 capital market line, 12 causal strength modeling, 42 causal strength modeling (CSM), 51–2 CDVine, 185 CLIME, 110, 111 co-monotone sets, 208 coherent risk measure, 260 cointegration-based trading strategies, 23 collateralized debt obligation (CDO), 279 conditional sparsity, 102 conditional value at risk (CVaR), 233, 234–42 minimization, 240–2, 263 support vector machines, 247–52 Financial Signal Processing and Machine Learning, First Edition Edited by Ali N Akansu, Sanjeev R Kulkarni and Dmitry Malioutov © 2016 John Wiley & Sons, Ltd Published 2016 by John Wiley & Sons, Ltd Index 290 conditional value at risk (CVaR) (continued) portfolio selection, 241–2 as risk measure, 263 robust optimization, 259–61 under finite scenarios, 236–8 under normal distribution, 235 constant relative risk aversion (CRRA), 159 constant shift insensitivity, 269 contract asymptotics, 141–2 convex relaxation, 16 copula modeling, 1, 179–85 Archimedean, 181, 182 multiple variables, 183–5 parametric, 181–3 product copula, 181 software, 185 copula (software), 185 correlation measures, 164–5 copulas, 179–85 fitting, 180–1 dependence types, 185–6 positive and negative, 185–6 tail, 187–8 Granger causality, 176–8 Huber-type, 166 Kendall’s tau, 166–7 misspecification testing ARCH effects, 172–4 Ljung-Box, 171 multivariate, 176 multiple variables, 183–5 Spearman’s rho, 168–9, 179, 182 covariance matrix estimation, 100–2 factor analysis, 100–8 asymptotic results, 105–7 example, 107–8 threshold, 105 unknown factors, 104–5 pure factor models, 126–7 optimal weight matrix, 129–30 covariance selection, 109 credit default swaps, 279 CRSP database, 124–6 CVar see conditional value at risk Dantzig selector, 111 dependence, 185–7 deviation, 270 Dirichlet distribution, 58 discrete cosine transform (DCT), 67, 71, 72 discrete Fourier transform (DFT), 67 domain description, 245–6 dynamic programming equation, 155–6 efficient frontier, 12 efficient market hypothesis (EMH), Eigen decomposition see principal components analysis EJD algorithm, 215, 219–20 error decomposition, 274 error measures, 269 error projection, 269 essential infimum, 268 essential supremum, 268 estimator’s breakdown point, 284–5 ETF see exchange-traded funds E𝑣-SVC, 248–9 exchange-traded funds, 42 expectation-maximization, 57 expected shortfall, 263 extreme joint distribution (EJD) algorithm, 215, 219–20 factor analysis, 100, 100–8, 120 covariance matrix estimation asymptotic results, 105–6 threshold, 105 unknown factors, 104 factor-pricing model, 121–2 fallout, 286 Fama-French model, 107–8 fixed income instruments, fixed strike price, 136 fixed transforms, 67 Frank copula, 182 Frechet-Hoeffding theorem, 215–17 Fused-DBN, 62, 63 GARCH model, 141, 172–4 BEKK GARCH model, 175 VECH GARCH, 175 Index Gauss-Markov theorem, 268 Gaussian copula, 181 genomic networks, 118–19 GICS, 115–16 Global Financial Crisis, 43 Global Industry Classification Standard (GICS), 115–16 global minimum variance portfolio, 120–1 Granger causality, 46–7, 176–7 nonlinear, 177–9 graphical Dantzig selector, 110 Green’s function, 143 group lasso, 20, 47 group OMP, 49, 51–2 H-J test, 178 high-density region estimation, 245–6 high-frequency trading, hinge loss, 244 Huber-type correlations, 166 implied volatility asymptotics, 143–5 implied volatility skew, 136 index tracking, 19 inference function for margins (IFM) method, 180 intensity randomization, 196–7 intercept, 267 interior point (IP), 53 interior point (IP method), 53 inverse projection, 270 investment risk, iShares, 53–5 Japan, 43, 55 jump-diffusion processes, 191–2 Karhunen-Loeve transform (KLT), 67–8 kernel derivation, 72–9 continuous process with exponential autocorrelation, 74–6 eigenanalysis of discrete AR(1) process, 76–7 fast derivation, 79–82 NASDAQ-100 index, 93–7 291 subspace sparsity, 82–4 pdf-optimized midtread reader, 84–6 see also principal components analysis Karhunen-Loeve expansion, 76 Kendall’s tau, 166–7, 169, 179, 182 kernel trick, 242 Kolmogorov backward equation (KBE), 172 Landweber’s iteration, 15 lasso regression, 17, 46, 252 adaptive, 47, 48, 50, 51 group, 47, 48, 57 SQRT, 112–13 least absolute deviation (LAD), 53 least median of squares (LMS) regression, 285 least-squares methods, 14, 268 ordinary least squares, 103, 251 POET and, 104 regularized, 48 temporal causal modeling, 48 least-trimmed-squares (LTS) regression, 285 leverage effect, 140 Lévy kernel, 150–1 linear regression, 283–4 liquidity, Ljung-Box test, 171 local volatility models, 139 local-stochastic volatility (LSV) models, 140–1, 146–8, 155–7 Heston, 148–50, 149 Lévy-type, 150–2 market incompleteness, 139 market microstructure, 4–5 market price of risk, 153 market risk see investment risk Markov-switched TCM, 60–4 Markowitz bullet, 12 Markowitz portfolio selection, 1, elastic net strategy, 19 as inverse problem, 13–15 portfolio description, 11–13 as regression problem, 13 sparse, 15–17 Index 292 Markowitz portfolio selection (continued) empirical validation, 17–18 optimal forecast combination, 20–1 portfolio rebalancing, 18 portfolio replication, 19 see also sparse Markowitz portfolios matrix deflation, 30 maximum a posteriori (MAP) modeling, 56, 58–9 mean-absolute deviation, 277–8, 279 mean-reverting portfolios, 24, 29 crossing statistics, 28, 31 mean-reversion proxies, 25–6 numerical experiments, 32–9 basket estimators, 33 historical data, 32–3 Jurek and Yang strategy, 33, 36, 37 Sharpe ratio robustness, 36–7 tradeoffs, 38–9, 38 transaction costs, 33–4 optimal baskets, 28–9 portmanteau criterion, 27–8, 29, 31 predictability multivariate case, 26–7 univariate case, 26 semidefinite relaxations, 30–1 portmanteau, 31 predictability, 30–1 volatility and sparsity, 24–5 mean-variance efficiency, 122 Merton problem, 155–60 mevalonate (MVA) pathway, 118 misspecification testing ARCH/GARCH, 172–4 Ljung-Box test, 171 multivariate, 176 model asymptotics, 142 monotone distributions, 208–14, 238–9 mortgage pipeline risk, 286 MSCI Japan Index, 42 NASDAQ-100, 93–7 negative dependence, 185–6 news analysis, Newton method, 53 no-arbitrage pricing, 139 no-short portfolios, 16–17 nonnegative space PCA, 83 nonnegative sparse PCA, 83 optimal order execution, optimized certainty equivalent (OCE), 264 option pricing asymptotic expansions, 141–2 contract, 142 implied volatility, 143–5 model, 142–3 model coefficient expansions, 146–50 model tractability, 145–6 Oracle property, 48 ordinary least square (OLS), 103, 251 Ornstein–Uhlenbeck (OU) process, 68 orthogonal patching pursuit (OMP), 49, 51–2, 54 outlier detection, 245–6 panel data models, 127–9 partial integro-differential equation (PIDE), 141 Pearson correlation coefficient, 162 penalized matrix decomposition, 83 penalties relative to expectation, 269 pension funds, perturbation theory, 142–3 POET, 104, 120–1 Poisson processes, 193–6 backward simulation (BS), 222–7 common shock model, 196 extreme joint distributions, 207–19 approximation, 217–19 Frechet-Hoeffding theorem, 215–17 monotone, 208–14 optimization problem, 207–8 intensity randomization, 196–7 numerical results, 219–22 simulation backward, 200–6 forward, 197–9 model calibration, 206 Poisson random vectors, 205 Poisson-Wiener process, 222–7 portfolio manager, Index portfolio optimization, 1=N puzzle, 13 Markowitz, 11–12 portfolio rebalancing, 18–19 portfolio risk estimation, 119–21 positive dependence, 185–6 positive homogeneity, 239 power enhancement test, 123–4 precision matrix estimation, 109–10 applications, 115–17 column-wise, 110–11 portfolio risk assessment, 119–26 TIGER, 112–14 application, 115–17 computation, 114 genomic network application, 118–19 theoretical properties, 114–15 tuning-insensitive procedures, 111–12 price inefficiency, principal components analysis (PCA), 34 principal orthogonal complement, 104 principal orthogonal complement thresholding (POET) estimator, 104 principal components analysis (PCA), 36–7, 67, 126 discrete autoregressive (AR(1)) model, 68–70 fast kernel derivation, 79–82 Eigen subspace, 70–1, 91–2 Eigen subspace sparsity, 82–3 KLT kernel continuous process with exponential autocorrelation, 74 eigenanalysis of a discrete AR(1) process, 76–9 orthogonal subspace, 69–72 Eigen subspace, 70–2 performance metrics, 71 pure factor models, 126–7 sparse methods, 83–4 AR(1), 89–91 Eigen subspace quantization, 86 Eigenvector pdf, 87–9 pdf-optimized midtread quantizer, 84–6 performance, 91–3 293 pulse code modulation (PCM), 71 pure factor model, 130 Q-TCM, 52 quadratic penalty, 14 quantile regression, 251 quantile TCM, 52–5, 54–5 quasi–Monte Carlo (QMC) algorithms, 203 reduced convex hulls, 257, 258 regressant, 267 regression analysis, 267–8 CVar-based, 251–3 error decomposition, 273–4 error and deviation measures, 268–71 lasso see lasso regression least-squares, regularized, 48 least-squares methods, 275–7 linear regression, 283–4 median regression, 277–81 ordinary least squares, 251 quantile regression, 281–3 risk envelopes and identifiers, 271–3 robust, 284–6 support vector regression, 246–7 regressors, 267 return on investment (ROI), 2–3 ridge regression, 251 risk acceptable linear regression, 284 risk envelopes, 271–3 risk inference, 121 risk preferences, 268 risk quadrangle, 264 risk-neutrality, 139 risk-normalized return, 2–3 robust optimization, distributional, 259–61 robust regression, 284–6 SCoTLASS, 83 SDP see semidefinite programs SDP relaxations for sparse PCA (DSPCA), 83 securities, 4, 11, 135 securities markets, semidefinite programs, 30 relaxation tightness, 31–2 Index 294 Sharpe ratio, 2–3, 36–7 transaction costs and, 38–9 Sherman-Morrison-Woodbury formula, 103 shift operator, 145 shortselling, signal-to-quantization-noise ratio (SQNR), 85 Sklar’s theorem, 179–80 soft convex hulls, 257 soft thresholding, 84 sparse KLT, 89–91 sparse Markowitz portfolios, 15–17 empirical validation, 17–18 portfolio rebalancing, 18–19 sparse modeling, 5–6 sparse PCA via regularized SVD (sPCA–rSVD), 83 sparse vector autoregressive (VAR) models, 42 Spearman’s rho, 168–9, 169–70, 179, 182 SQRT-lasso, 112–13 stationary sequences, 165 statistical approximation theory, 268 statistical arbitrage, stochastic volatility, 1, 135 Black-Scholes model, 136–7 dynamic programmic equation, 155–7 implied volatility, 137 local volatility models, 139–40 Merton problem, 155–60 separation of timescales approach, 152–3 stochastic volatility models, 140 local (LSV), 140–1 with jumps, 141 volatility modeling, 137–41 volatility of volatility, 152 stock exchanges, stock return analysis, 62–3, 64 Strong Law of Large Numbers, 192, 212 support vector machines (SVM), 233–4, 263 classification, 242–3 C-support, 243–4 duality, 256–9 soft-margin, 244–5 𝑣-SVM, 247 geometric interpretation, 257–9 support vector regression (SVR), 246–7 Survey of Professional Forecasters (SPF), 21 Switzerland, 43 tail dependence, 187–8 tail VaR, 263 temporal causal modeling (TCM), 42, 44–5 algorithmi overview, 47 Bayesian group lasso, 57–8 extensions causal strength modeling, 51–2 quantile TCM, 52–5 Granger causality and, 46–7 grouped method, 47–9 greedy, 49 regularized least-squares, 48 Markov switching model, 56–7 stock return analysis, 62–3 synthetic experiments, 60–2 maximum a posteriori (MAP) modeling, 58–9 quantile TCM, 54–5 regime change identification, 55–63 algorithm, 58–60 synthetic experiments, 49–50, 60–2 data generation, 49–50, 60–1 TIGER, 112–14 computation, 114 Tikhonov regularization, 14 time series, 165 Tobin two-fund separation theorem, 12 transcendental equations, 73–4 transform coding, 71 translation invariance, 239 truncated singular value decomposition (TSVD), 15 tuning-insensitivity, 113 TV-DBN, 61, 62 two-tailed [alpha]-value-at-risk (VaR) deviation, 286 unbiased linear regression, 283–4 Index v-property, 253–5 𝑣-SVM, 257–8 𝑣-SVR, 247, 251–2 value function, 156 value-at-risk, 234 Vapnik–Chervonenkis theory, 243 VECH GARDCH model, 175 vector autoregression (VAR), 25, 49–50, 61, 174 multivariate volatility, 175–6 temporal causal modeling, 57–8, 61–2 295 VineCopula, 185 volatility, 38–9 volatility of volatility modeling, 152 Wald test, 122–3 weighted principal components (WPC), 127, 130–1 Yahoo! Finance, 115–17 zero crossing rate, 28 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... FINANCIAL SIGNAL PROCESSING AND MACHINE LEARNING FINANCIAL SIGNAL PROCESSING AND MACHINE LEARNING Edited by Ali N Akansu New Jersey Institute... the Black–Scholes Financial Signal Processing and Machine Learning formulation and the notion of implied volatility, and discusses local and stochastic models of volatility and their asymptotic... financial engineering: financial signal processing and electronic trading New York: Academic-Elsevier 10 Financial Signal Processing and Machine Learning Almgren, R and Chriss, N (2001) Optimal